logistic analysis of channel pattern thresholds:...

Ž .Geomorphology 38 2001 281–300www.elsevier.nlrlocatergeomorph

Logistic analysis of channel pattern thresholds: meandering,braiding, and incising

Brian P. Bledsoe), Chester C. Watson 1

Department of CiÕil Engineering, Colorado State UniÕersity, Fort Collins, CO 80523, USA

Received 22 April 2000; received in revised form 10 October 2000; accepted 8 November 2000

Abstract

A large and geographically diverse data set consisting of meandering, braiding, incising, and post-incision equilibriumstreams was used in conjunction with logistic regression analysis to develop a probabilistic approach to predicting thresholdsof channel pattern and instability. An energy-based index was developed for estimating the risk of channel instabilityassociated with specific stream power relative to sedimentary characteristics. The strong significance of the 74 statisticalmodels examined suggests that logistic regression analysis is an appropriate and effective technique for associating basichydraulic data with various channel forms. The probabilistic diagrams resulting from these analyses depict a more realisticassessment of the uncertainty associated with previously identified thresholds of channel form and instability and provide ameans of gauging channel sensitivity to changes in controlling variables. q 2001 Elsevier Science B.V. All rights reserved.

Keywords: Channel stability; Braiding; Incision; Stream power; Logistic regression

1. Introduction

Excess stream power may result in a transitionfrom a meandering channel to a braiding or incising

Žchannel that is characteristically unstable Schumm,.1977; Werritty, 1997 . Changes in the magnitude,

relative proportions, and timing of sediment andwater delivery in these channels may affect waterquality via a wide variety of mechanisms. Thesemechanisms include changes in channel morphology,planform, bed material, and suspended sediment

) Corresponding author. Tel.: q1-970-491-8410; fax: q1-970-491-8671.

E-mail addresses: [email protected]Ž . Ž .B.P. Bledsoe , [email protected] C.C. Watson .

1 Tel.: q1-970-491-8313; fax: q1-970-491-8671.

loads, loss of riparian habitat because of stream bankerosion, and changes in the predictability and vari-ability of flow and sediment transport characteristics

Ž .relative to aquatic life cycles Waters, 1995 . Inaddition, braiding and incising channels frequentlythreaten human infrastructure.

Alluvial channel patterns form a continuum ratherthan discrete types and distinctions between morpho-

Žlogic types are fuzzy and complex Ferguson, 1987;.Knighton and Nanson, 1993 . Process and historical

studies of individual streams suggest that switchesbetween channel patterns may be historically contin-gent on how intrinsic variables have ‘primed’ areach for instability and on the state of the channel at

Ž .the time of an impact Brewer and Lewin, 1998 .Because channel patterns are frequently transitionaland shaped by complex sequences of disturbance

0169-555Xr01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved.Ž .PII: S0169-555X 00 00099-4

( )B.P. Bledsoe, C.C. WatsonrGeomorphology 38 2001 281–300282

events, stochastic alternatives to deterministic thresh-olds may be more useful and realistic for predictiveand postdictive work on the effects of geomorphic

Ž .change Graf, 1981; Tung, 1985; Ferguson, 1987 .Probabilistic discrimination of relatively stable andunstable channel types using basic sedimentary andhydraulic variables offers improved predictions ofenvironmental impacts. Such predictions of risk helpestablish priorities for management, planning, andpolicy.

1.1. Discriminators of braiding and meandering

In the latter half of the twentieth century, geomor-phologists and engineers attempted to predict theplanform of rivers based on the slope and somemeasure of discharge or drainage area. The concept

Ž .of a threshold discharge–slope Q–S combinationthat discriminates braiding rivers from meanderingones has become firmly embedded in the doctrine of

Ž .fluvial geomorphology Carson, 1984 . The Q–SŽ .discriminator was first suggested by Lane 1957 and

Ž .Leopold and Wolman 1957 . Many other investiga-tors have provided support for the notion of a thresh-old between meandering and braiding types and haveused this approach to interpret a variety of field and

Žflume data Schumm and Khan, 1972; Chitale, 1973;.Osterkamp, 1978; Begin, 1981; van den Berg, 1995 .

Critical examination of the early work on mean-dering to braiding thresholds has subsequently re-vealed several limitations of the original approach.Critical stream power for braiding appears to varywith bed material size; i.e., the critical slope forbraiding at a given discharge is higher for gravel

Žthan for sand bed channels Carson, 1984; Ferguson,.1987 . The threshold suggested by Leopold and Wol-Ž .man 1957 is too high for sand bed channels and too

low for gravel bed channels because it is based on acombination of the two channel types. Other con-cerns with the original stream power approach arisefrom bias associated with using channel slope and

Ž .bankfull discharge parameters van den Berg, 1995 .Channel slope and bankfull discharge are not neces-sarily independent of planform. Braided channels areinherently steeper than meandering channels for agiven valley slope because of generally low sinuos-ity. Braided channels may also have larger bankfullcross-sections as a consequence of braiding. Some

investigators have also suggested that bank resis-tance is a confounding factor in the analysis of

Žchannel planform Carson, 1984; Ferguson, 1987;.Bridge, 1993 .

Several theoretical approaches to defining thetransition from meandering to braiding have also

Žbeen developed e.g., Parker, 1976; Fredsøe, 1978;.Fukuoka, 1989 . These techniques have limited pre-

dictive value, however, because the techniques relyon parameters such as width, depth, and Froudenumber that are usually not available for a prioripredictions of channel form. The theoretical ap-proaches do seem to suggest that braiding generallyoccurs at channel width-to-depth ratios in excess of

Ž . Ž .50 Fredsøe, 1978 . Bridge 1993 provides an excel-lent review and summary of the various empiricaland theoretical thresholds that have been proposedfor the transition from braiding to meandering.

In an attempt to reconcile some of the issuesassociated with the original empirical approaches,

Ž .van den Berg 1995 devised a simple parameterrepresenting potential specific stream power that en-ables discrimination of braiding and meandering

Ž .rivers sinuosityG1.3 in unconfined alluvium. Us-ing a data set of 126 streams and rivers, he arrived ata discriminant function of the form:

g0.5 0.42v( S Q s900D 1Ž .

Õ bf 50a

Ž .where specific stream power v is a function ofŽ 3 .valley slope, estimated bankfull discharge m rs ,

Ž .and a regression coefficient a computed for aparticular collection of streams that varies betweensand bed and gravel bed rivers. The discriminatefunction applies to both sand and gravel bed riverswith median bed material ranging from 0.1 to 100mm. Potential specific stream power and median

Ž .particle size D are expressed in Watts per square50Ž 2 .meter Wrm and meters, respectively. The result-

ing specific stream power represents the maximumpotential specific stream power for a particular val-ley slope.

1.2. Stream power and eÕolution of incising sand bedstreams

Channel incision, or bed lowering by erosion,results from an imbalance in the power aÕailable to

( )B.P. Bledsoe, C.C. WatsonrGeomorphology 38 2001 281–300 283

move a sediment load and the power needed toŽ .move the sediment load Bull, 1979; Galay, 1983 .

Where banks are cohesive and resistant to hydraulicforces, bed erosion is initially dominant over channelwidening. As an incising channel deepens, flowevents of increasingly greater magnitudes are con-tained within the channel banks. This process maycreate a positive feedback wherein shear stress onthe bed and toe of the banks during large events ismagnified through incision. If bed degradation con-tinues until banks reach a critical height for geotech-nical failure, radical channel widening may be initi-ated. Such widening can proceed at a staggering rate,with tenfold increases in cross-sectional areas re-

Ž .ported in some instances Harvey and Watson, 1986 .Conceptual models of the temporal response or

evolutionary sequence of incised channels have beenŽdescribed by various investigators Schumm et al.,

1984; Harvey and Watson, 1986; Watson et al.,.1988a,b; Simon, 1989; Simon and Hupp, 1992 . The

models generally describe the same process: excessspecific stream power initially results in bed degra-dation followed by a period of widening and bankfailure and further bed adjustment until the channeleventually establishes a new flood plain and quasi-equilibrium channel form within newly depositedalluvium.

When compared with work on the meandering tobraiding threshold, incised channel forms have re-ceived relatively little attention from researchers.Even less common are investigations of thresholds ofstream power associated with the development ofincised channel forms.

1.3. ObjectiÕes

Water resources engineers and fluvial geomor-phologists sometimes need to relate qualitative de-pendent variables to one or more independent vari-ables, which may or may not be quantitative. In thecase of channel instability and ecosystem changesbecause of excess stream power, the dependent vari-able might be the occurrence or nonoccurrence of arapid and significant geomorphic response in theform of widening andror incision. Thus, the depen-dent variable is binary and qualitative and the inde-pendent variable may be quantitative and continuous,

such as specific stream power, the size of bed mate-rial, sediment load, and bank resistance. Logistic

Ž .regression Menard, 1995; Christensen, 1997 is astatistical technique that allows comparison of binaryand continuous parameters. The research describedbelow is targeted at developing a risk-based ap-proach for predicting the occurrence of stable mean-dering, braiding, and incising channel forms as afunction of simple hydraulic and sedimentary at-tributes. The specific objectives of the proposedresearch were the following.

Ž .i To use a very large data set from actualstreams with logistic regression methodology toidentify probabilities of occurrence of braiding orincising channels versus meandering channels usingsimple indices of specific stream power and charac-teristics of bed material.

Ž .ii To contrast the logistic regression results withŽ .van den Berg’s 1995 previously defined geomor-

phic threshold between meandering and braidingforms.

2. Methods

2.1. Logistic regression analysis— basic concepts

Commonly applied regression techniques are ap-propriate only when the dependent variable and theexplanatory variables are quantitative and continu-

Ž .ous. To analyze a binary qualitative variable 0 or 1as a function of a number of explanatory variables,special techniques must be used if the analysis is tobe performed adequately. The regression problemmay be revised so that, rather than predicting abinary variable, the regression model predicts a con-tinuous variable that stays within the 0–1 bounds.One of the most common regression models thataccomplishes this is the logit or logistic regression

Ž .model Menard, 1995; Christensen, 1997 . Althoughlogistic regression is most frequently applied in the

Ž .social and health sciences, Tung 1985 presented alogistic regression methodology to evaluate the po-tential of channel scouring as a function of quantita-tive data taken from flume studies.

In the logistic regression model, the predictedvalues for the dependent variable are never F0 orG1 regardless of the values of the independent


variables. This is accomplished by applying the fol-lowing regression model:

exp b qb x q . . .qB xŽ .0 1 1 n nys 2Ž .

1qexp b qb x q . . .qb xŽ .0 1 1 n n

Regardless of the regression coefficients or the mag-nitude of the x values, this model will alwaysproduce predicted values in the range of 0 to 1.Suppose the binary dependent variable y representsan underlying continuous probability p, ranging from0 to 1. The probability p may be transformed as:

pXp s ln 3Ž .ž /1yp

This transformation is referred to as the logit orlogistic transformation. Note that pX can theoreticallyassume any value between minus and plus infinity.Since the logit transform solves the issue of the 0–1

Žboundaries for the original dependent variable prob-.ability , the logit transformed values could be used in

Ž .an ordinary linear regression equation. If Eq. 3 isapplied to both sides of the logistic regression equa-

Ž .tion stated in Eq. 2 , the standard linear regressionmodel is obtained:

pXsb qb x q . . .qb x 4Ž .0 1 1 n n

In logistic regression analysis, maximum likeli-hood techniques may be used to maximize the valueof a function, the log-likelihood function, whichindicates how likely it is to obtain the observedvalues of the dependent variable given the values ofthe independent variables and parameters, b , . . . ,b .0 n

Unlike ordinary least squares regression, which isable to solve directly for parameters, the solution tothe logistic regression model is found through aniterative process that maximizes the likelihood func-tion until the solution converges within tolerance.Maximum likelihood estimation was used for alllogistic regression analyses described in this study. Aquasi-Newton method provided in the statistical

w Ž w .analysis software Statistica StatSoft , 1999 wasused to optimize the maximum likelihood function.Model results from Statisticaw were verified using

Ž .the SAS software package SAS Institute, 1990 .

2.2. Regression interpretation and diagnostics

In ordinary regression analysis, the coefficient ofŽ 2 .determination R is frequently used as a measure

of model performance. In logistic regression, it iscommon to be more concerned with whether thepredictions are correct or incorrect than with how

Žclose the predicted values are to the observed 0 or. 21 values of the dependent variables. Therefore, R

has little meaning in logistic regression analysisŽ .Tung, 1985; Christensen, 1997 . One of the mostmeaningful ways of interpreting a logistic regressionmodel is to examine the number of incidences of

Ž .misclassification Tung, 1985 . The number of cor-rect and incorrect classifications may be used to

Ž .calculate an ‘odds ratio’ Neter et al., 1988 from a2=2 classification table which displays the pre-dicted and observed classification of cases for abinary dependent variable. Although the odds ratioprovides a useful measure of predictive accuracy, itis quite sensitive to sample size. Care must be takenin comparing across models that have differing num-bers and proportions of dependent variable cases. Inthis study, the percentages of streams and riverscorrectly classified by the various logistic regressionmodels was used as a primary measure of modelperformance.

Goodness-of-fit tests may also aid in the interpre-tation of the results of logistic regression. The likeli-hood L for the null model, where all slope parame-0

ters are zero, may be directly compared with thelikelihood L of the fitted model. Specifically, one1

can compute the x 2 statistic for this comparison as:

x 2sy2 log L y log L 5Ž . Ž . Ž .Ž .0 1

The degrees of freedom for this x 2 value are equalto the number of independent variables in the logisticregression. If the p-level associated with this x 2 issignificant, the estimated model yields a significantlybetter fit to the data than the null model and theregression parameters are statistically significant.This statistic is equivalent to the F statistic in ordi-nary least squares regression.

Logistic regression diagnostics, like using linearregression diagnostics, may seem more art than sci-

Ž .ence Menard, 1995 . Diagnostic statistics hint atpotential problems, but whether remedial action isrequired is a matter of judgement after close inspec-tion of the data. In a sample of 200–250, randomsampling variation alone will produce 10–12 casesof residuals with absolute values )2 on standard-ized, normally distributed variables such as the de-


viance residual or the Studentized residual. Verylarge residuals do not necessarily indicate problems

Ž .in the model Menard, 1995 . As described below,only a few independent variables were consideredfor inclusion in the logistic regression models ofchannel response. Measures of flow energy and sedi-mentary characteristics were either combined as asingle parameter or used as two separate, indepen-dent variables. Therefore, issues associated withmodel mis-specification and multi-collinearity of in-dependent variables were kept to a minimum. Theresiduals of each regression model were examinedfor outliers, influential cases, and non-normality. Asmight be expected with a large and diverse data seton river characteristics, some rivers were consis-tently misclassified. Such cases were examined forpotential errors, but all cases were retained in thedata set because no compelling reasons existed tosuspect that mistakes in reporting or measurementhad occurred.

2.3. Field data used in analyses of channel patternand stability

A primary objective of this study was to developan extensive data set on slope, discharge, character-istics of bed material, planform, and morphology ofalluvial rivers from various regions of the world.Data chosen for the analysis were selected fromseveral scholarly publications including peer-re-viewed articles from scientific journals and govern-ment reports and documents. Streams and riversincluded in the data set ranged from live bed chan-nels with beds of very fine sand to threshold chan-nels composed of cobble-sized materials. The datawere limited to self-formed channels in alluvial ma-terial that are free from constraints such as bedrockoutcrops and stabilization structures that restrict theevolution of braided rivers. The resulting data set of270 streams and rivers from around the world alongwith sources can be obtained from the correspondingauthor.

Five categories of channels were delineated forthe analyses: stable meandering, braiding, incising,braiding or incising, and quasi-equilibrium channels.Stable meandering channels were defined as single-thread channels that have maintained a sinuositygreater than or equal to 1.3 without progressive

aggradation, degradation, or changes of width for aperiod of years. Streams exhibiting dynamic stabil-ity, such as lateral migration without significantchanges in width and depth, were deemed acceptablefor inclusion. Like meandering, braiding has beendefined in numerous ways by various investigators.In this analysis, braiding referred to multiple-threadchannels with generally unvegetated bars and islandsthat were not significantly wider than the adjacentchannels. Unfortunately, insufficient data precludedattempts to distinguish between actively braiding andrelatively inactive braided streams.

By contrast, only streams that were actively andprogressively incising over periods of 1–10 yearswere included in the analysis. Channels identified asincising were all classified as CEM Type 2 accord-

Ž .ing to the Watson et al. 1988a,b model of incisedchannel evolution. Type 2 channels are in the earlystages of rapid incision and have yet to initiate thedramatic widening that occurs in subsequent stagesof evolution. Quasi-equilibrium channels were de-fined as channels that have a stable slope after a full

Ž .cycle of incision CEM Types 4 and 5 according toŽ .the Watson et al. 1988a,b model of channel evolu-

tion.To facilitate the separation of channels with sand

beds from the rest of the data set during statisticalanalyses, the data were also tagged as having one ofthree categories of bed material: sand, mixed, orgravel. Sand was defined as a D from 0.0625–250

mm, mixed was defined as )2–10 mm, and gravelwas defined as )10 mm. These distinctions areadmittedly arbitrary, but such a separation allows arough discrimination between live bed channels thatare very sensitive to sediment supply and threshold

Žchannels that are somewhat less sensitive Mont-. Ž .gomery and Buffington, 1997 . van den Berg 1995

reported that relatively few streams occur with a D50

between 2 and 10 mm. The mixed channels wereonly included in collective analyses of all size frac-tions.

2.4. Variables used in the analyses

The parameters considered for inclusion in thelogistic regression models were limited to the basicvariables of slope, discharge, and some representa-tion of bed material. Sinuosity was used to relate bed


slope to valley slope where valley slope data werenot available. In cases where sinuosity data were notavailable for braided rivers, valley slope was as-sumed to be 10% steeper than bed slope. Bed slopewas assumed to be representative of energy slope forall streams with the exception of incising and quasi-equilibrium streams in the Yazoo River watershed ofnorthern Mississippi. For these streams, energy slopes

Žwere estimated using surveyed cross-sections Wat-.son et al., 1998a, 1999 in the hydraulic model

Ž .HEC-RAS USACE, 1997 .Unfortunately, limiting the analysis to streams

with detailed descriptions of the characteristics ofbed materials would have drastically reduced the sizeof the data set. For a great number of streams, only

Ž .the median particle size D was reported. As a50

result, only the diameter of the median particle sizewas utilized to maximize sample size. Estimates ofthe median size of bed materials were the result ofsieve analyses for sand fractions and Wolman countsŽ .Wolman, 1954 for gravel and coarser materials.Sediment information was not available for somestreams in the Yalobusha River watershed in Missis-sippi. Median size of sediment in the basin, however,is extremely uniform within a range of 0.3–0.5 mmŽ .Watson et al., 1998b . The median size of particlesin unsampled streams was assumed to equal themedian size of 0.36 mm for streams that have beensampled in the Yalobusha watershed.

2.5. DeÕelopment of model parameters

In developing parameters for the logistic regres-sion models, emphasis was placed on maintainingindependent variables in forms that clearly representphysical processes having a theoretical foundation.

Ž . Ž .Specifically, slope S and discharge Q were com-bined into one variable representing a surrogate forspecific stream power. A preliminary logistic regres-sion analysis was performed using combinations ofthis parameter in conjunction with D and fall50

velocity. The results of this analysis suggested that

QS 6Ž .(D50

is a simple but robust predictor of channel planformand stability. In some cases, this single parameter

had explanatory power equaling that of models con-taining slope, discharge, and D as separate inde-50

pendent variables. This was especially true for chan-nels with sand beds. The inclusion of D in the50

denominator simply scales flow energy relative tothe size of bed material. The form of this parameteris also consistent with the theoretical relationships

Ž .developed by Chang 1985 and the empirical rela-Ž .tionship of van den Berg 1995 . This parameter is

approximately proportional to unit stream power, VSŽ .Yang, 1976 , multiplied by a relative submergenceterm:

Q S Re hS A AVS 7Ž .0.5( ž /D DD(50 5050

The ratio of forces represented by this parametersuggests that it is an index of erosive power ormobility. Henceforth, it will be referred to as a

Ž .Amobility index.B It is worth noting that Eq. 6 maybe made dimensionless by adding kinematic viscos-

Ž .ity Õ to the denominator under the radical:

QS 8Ž .(D Õ50

If one assumes a value of 10y6 m2rs for clear waterŽ .at 208C, then the value of Eq. 8 is simply the value

Ž . 3of Eq. 6 multiplied by a factor of 10 in SI units.The dimensionless value of this parameter is notreported in this study because potential differences inkinematic viscosity make application and interpreta-tion more difficult.

Ž .The mobility index represented in Eq. 6 wasused extensively in the logistic regression analyses.In addition, models containing SQ0.5 with D as50

separate independent variables were employed. Anal-yses were performed using bedslope and valley slope.Annual flood and bankfull discharges were utilizedin combination and separately to allow a comparisonof accuracy. The log transforms of all data were10

utilized in the logistic regression models. Two-parameter logistic regression models were of theform:

exp b qb x qb xŽ .0 1 1 2 2p instability s 9Ž . Ž .

1qexp b qb x qb xŽ .0 1 1 2 2


where x is the specific stream power term; x is1 2Ž .the median bed material term; and p instability is

the probability of braided, incising, or unstable chan-nel types when regressed against stable meanderingor quasi-equilibrium channels. The single parameter

Ž .models were equivalent to Eq. 9 without the b x2 2

terms. In the single parameter models, the mobilityŽ Ž ..index Eq. 6 was used exclusively as the indepen-

dent variable. Over 200 logistic regression modelswere tested using one- and two-parameter ap-proaches. The logistic regression results providedexplicit probabilistic statements that were attached todiagrams of channel stability to clearly associate therisk of a particular channel form with varying levelsof excess stream power.

2.6. Relating logistic regression results to otherthresholds

The logistic regression results were compared withthresholds of channel form and stability reported by

Ž .van den Berg 1995 . An interesting feature of logis-tic regression is the potential for transforming a

Ž .relationship like Eq. 9 into a function representingthe combination of independent variables corre-sponding to a 50% chance of the binary-dependentvariable occurring in either category. For example,assume that a two-parameter logistic regressionmodel has been fit to log transformed data on10

meandering and braiding streams. The combinationof independent variables resulting in a probability ofbraiding of 0.5 represents the circumstances underwhich it is most uncertain whether a stream will bebraiding or meandering. Recalling that for a two-parameter logistic model:

pln sb qb x qb x 10Ž .0 1 1 2 2ž /1yp

it follows that the combination of independent vari-ables corresponding to ps0.5 chance of meander-ing or braiding may be written as:

b b0 2x sy y x 11Ž .1 2

b b1 1

Assuming that x and x happen to be log SQ0.51 2 10

and log D , respectively, then a power function10 50

approximately corresponding to ps0.5 is:

0 2b by y'S Q s10 D 12Ž .1 1b b50

This approach was used in comparisons with theŽ .discrete threshold proposed by van den Berg 1995 .Ž .In addition, the original data of van den Berg 1995

were re-analyzed using a two-parameter logistic re-gression model. This allows a risk-based interpreta-

Ž .tion of the discriminant function presented in Eq. 1 .

3. Results and discussion

A comprehensive summary of logistic regressionresults for 74 models is presented in Table 1. Eachlogistic regression model was assigned a number inTable 1 that is used in referring to specific modelsthroughout this discussion. The excellent fit of thesemodels underscores the predictive power of the sim-ple indices used to represent the balance of erosiveand resistant forces. All log-likelihood x 2 statisticsof the models were significant at the p-10y5 top-10y29 levels.

In general, the most accurate results for the great-est number of cases of all stream types were ob-tained by using bedslope in conjunction with annualflood discharge as first priority with bankfull dis-charge substituted for those streams without annualflood data. Overall, differences in predictive accu-racy, resulting from using combinations of annualflood and bankfull discharge data versus using an-nual flood or bankfull alone, were generally less than10%. The most notable exceptions were detectedwhen using valley slope instead of bedslope to repre-sent specific stream power. The predictive accuracyof the logistic regression models was reduced in allcases when bedslope was replaced with valley slope.

The one- and two-parameter models varied inexplanatory power depending on the type of chan-nels examined. For example, the single parameter

Ž .0.5model using the mobility index S QrD was50

extremely robust in analyses of channels with sandbeds using bedslope. In contrast, two-parameter

()

B.P

.Bledsoe,C

.C.W

atsonr

Geom

orphology38

2001281

–300

288

Table 1Summary of the results of logistic regression

a b 2 cŽ .Model no. Independent variable s Bed % Classified correctly b b b x Odds ratio n Flow used0 1 2

M B I BI Q0.5Ž Ž . .1 log S QrD srmrg 92.6 73.7 2.01 5.06 – 109.25 35.13 206 Q qQ10 50 af bf

2 sand 97.9 84.2 2.33 10.97 – 55.08 250.67 67 Q qQaf bf

3 gravel 94.6 73.5 3.75 6.50 – 76.38 48.89 127 Q qQaf bf0.5Ž Ž . .4 log S QrD srmrg 89.9 56.7 0.84 3.78 – 70.85 11.68 206 Q qQ10 v 50 af bf

5 sand 89.6 59.1 0.24 5.44 – 30.52 12.42 67 Q qQaf bf

6 gravel 93.5 64.7 2.45 5.69 – 58.26 26.58 127 Q qQaf bf

7 sand 95.8 63.6 y0.47 160.97 y57.85 33.30 40.25 67 Q qQaf bf

8 gravel 96.8 58.8 y2.06 39.91 y2.10 56.89 42.86 127 Q qQaf bf0.5Ž . Ž .9 log SQ and log D srmrg 94 75 11.87 6.54 y2.22 131.05 46.67 206 Q qQ10 10 50 af bf

10 sand 97.9 84.2 18.12 10.75 y6.06 55.11 250.67 67 Q qQaf bf

11 gravel 94.6 76.5 12.53 6.52 y2.65 79.58 57.2 127 Q qQaf bf0.5Ž . Ž .12 log S Q and log D srmrg 92.6 61.7 8.34 5.1 y1.69 86.04 20.18 206 Q qQ10 v 10 50 af bf

13 sand 95.8 68.4 8.53 6.48 y6.15 37.35 49.83 67 Q qQaf bf

14 gravel 93.5 67.6 10.35 5.71 y2.45 58.39 30.32 127 Q qQaf bf0.5Ž Ž . .15 log S QrD sand 89.6 92.9 4.63 14.39 – 82.45 111.80 90 Q qQ10 50 af bf

0.5Ž Ž . .16 log S QrD sand 72.9 71.4 1.38 6.29 – 35.40 6.73 90 Q qQ10 v 50 af bf0.5Ž . Ž .17 log SQ and log D sand 87.5 90.5 24.08 14.34 y12.12 84.43 66.50 90 Q qQ10 10 50 af bf0.5Ž . Ž .18 log S Q and log D sand 77.1 73.8 9.55 6.44 y6.97 39.59 9.48 90 Q qQ10 v 10 50 af bf

0.5Ž Ž . .19 log S QrD sand 85.4 91.8 4.51 13.15 – 100.06 65.60 109 Q qQ10 50 af bf0.5Ž Ž . .20 log S QrD sand 68.8 82 1.63 6.35 – 49.68 10.00 109 Q qQ10 v 50 af bf

0.5Ž . Ž .21 log SQ and log D sand 87.5 93.8 19.94 11.40 y8.83 98.39 105.00 109 Q qQ10 10 50 af bf0.5Ž . Ž .22 log S Q and log D sand 70.8 85.2 9.57 6.18 y6.23 53.36 14.03 109 Q qQ10 v 10 50 af bf

0.5Ž Ž . .23 log S QrD sand 88.1 68 4.28 13.95 – 43.18 15.73 67 Q qQ10 50 af bf0.5Ž . Ž .24 log SQ and log D sand 90.5 68 25.38 13.87 6.18 43.01 20.19 67 Q qQ10 10 50 af bf

0.5Ž Ž . .25 log S QrD srmrg 92.6 70.2 1.87 4.95 – 108.53 29.52 206 Q qQ10 50 bf af

26 sand 97.8 84.2 2.07 10.94 – 54.45 250.67 67 Q qQbf af

27 gravel 94.6 79.4 4.13 7.01 – 83.20 67.89 127 Q qQbf af0.5Ž Ž . .28 log S QrD srmrg 90.6 59.6 0.68 3.73 – 68.65 14.26 206 Q qQ10 v 50 bf af

29 sand 93.8 73.7 0.01 7.46 – 35.19 42.00 67 Q qQbf af

30 gravel 95.7 67.6 2.54 5.93 – 61.10 46.52 127 Q qQbf af0.5Ž . Ž .31 log SQ and log D srmrg 95.3 77.2 12.64 7.14 y2.18 134.99 68.66 206 Q qQ10 10 50 bf af

32 sand 97.9 84.2 17.61 10.66 y6.19 54.49 250.67 67 Q qQbf af

33 gravel 95.7 76.5 13.81 7.01 y3.01 83.33 72.31 127 Q qQbf af0.5Ž . Ž .34 log S Q and log D srmrg 94.6 64.9 8.96 5.64 y1.70 91.42 32.61 206 Q qQ10 v 10 50 bf af

35 sand 97.9 73.7 9.03 7 y6.71 37.15 131.6 67 Q qQbf af

36 gravel 95.7 64.7 11.00 5.93 y2.70 61.15 40.79 127 Q qQbf af0.5Ž Ž . .37 log S QrD sand 87.5 90.5 4.35 14.07 – 78.41 66.50 90 Q qQ10 50 bf af

0.5Ž Ž . .38 log S QrD sand 70.8 69 1.22 5.96 – 30.37 5.42 90 Q qQ10 v 50 bf af0.5Ž . Ž .39 log SQ and log D sand 83.3 90.5 23.93 13.9 y9.79 79 47.5 90 Q qQ10 10 50 bf af

()

B.P

.Bledsoe,C

.C.W

atsonr

Geom

orphology38

2001281

–300

289

Ž 0.5. Ž .40 log S Q and log D sand 70.8 69 8.72 5.81 y5.81 33.04 5.42 90 Q qQ10 v 10 50 bf af0.5Ž Ž . .41 log S QrD sand 85.4 91.8 4.21 12.77 – 95.62 65.60 109 Q qQ10 50 bf af

0.5Ž Ž . .42 log S QrD sand 66.7 80.3 1.46 6.07 – 44.41 8.17 109 Q qQ10 v 50 bf af0.5Ž . Ž .43 log SQ and log D sand 85.4 91.8 22.93 12.64 y6.81 95.65 65.6 109 Q qQ10 10 50 bf af0.5Ž . Ž .44 log S Q and log D sand 70.8 83.6 8.90 5.77 y5.57 47.25 12.39 109 Q qQ10 v 10 50 bf af

0.5Ž ( . .45 log S QrD sand 88.6 95.2 7.89 22.11 – 82.61 155.00 77 Q10 50 af0.5Ž Ž . .46 log S QrD sand 71.4 81 2.09 7.81 – 36.16 10.63 77 Q10 v 50 af

0.5Ž . Ž .47 log SQ and log D sand 94.3 95.2 37.5 21.43 y16.5 84.54 330 77 Q10 10 50 af0.5Ž . Ž .48 log S Q and log D sand 74.3 85.7 13.16 8.47 y8.62 40.69 17.33 77 Q10 v 10 50 af

0.5Ž Ž . .49 log S QrD sand 88.1 68 4.28 13.95 – 43.18 15.73 67 Q10 50 af0.5Ž . Ž .50 log SQ and log D sand 88.1 68 25.61 13.99 y6.22 43.20 15.73 67 Q10 10 50 af

0.5Ž ( . .51 log S QrD srmrg 91.7 77.4 2.10 4.98 – 105.28 37.93 174 Q10 50 bf

52 sand 96.2 88.9 2.50 13.40 – 42.97 200.00 44 Qbf

53 gravel 97.8 77.4 3.93 6.91 – 78.36 150.86 121 Qbf0.5Ž Ž . .54 log S QrD srmrg 90.1 66 1.06 4.09 – 74.61 17.66 174 Q10 v 50 bf

55 sand 84.6 77.8 0.34 8.00 – 28.56 19.25 44 Qbf

56 gravel 96.7 67.7 2.47 6.02 – 58.73 60.90 121 Qbf0.5Ž . Ž .57 log SQ and log D srmrg 95 81.1 12.03 6.72 y2.23 119.07 82.42 174 Q10 10 50 bf

58 sand 96.2 88.9 17.35 13.42 y17.98 45.46 200.00 44 Qbf

59 gravel 97.8 77.4 13.94 6.91 y3.24 78.38 150.86 121 Qbf0.5Ž . Ž .60 log S Q and log D srmrg 95 67.9 8.98 5.47 y1.89 84.69 40.59 174 Q10 v 10 50 bf

61 sand 84.6 88.9 8.96 8.31 y12.57 33.15 44.00 44 Qbf

62 gravel 96.7 67.7 11.42 6.02 y2.97 56.73 60.90 121 Qbf0.5Ž . Ž .63 log S Q and log D srmrg 93.1 76.9 16.21 10.06 y3.25 94.60 45.00 126 VDB Q10 v 10 50 bf

64 sand 94.4 80 12.80 10.42 y11.43 23.52 68.00 28 VDB Qbf

65 gravel 95.3 77.8 14.10 10.33 y1.82 66.91 71.17 91 VDB Qbf0.5Ž . Ž .66 log S Q and log D srmrg 89.8 80 10.46 6.83 y1.54 57.03 35.33 89 VDB Q10 v 10 50 af

67 sand 100 50 8.89 6.38 y3.47 7.02 – 39 VDB Qaf

68 gravel 71.4 84.6 9.48 6.86 y0.91 27.55 13.75 47 VDB Qaf0.5Ž . Ž .69 log S Q and log D srmrg 94.8 73.8 13.79 8.78 y2.54 98.3 51.20 157 VDB Q qQ10 v 10 50 bf af

70 sand 100 72.7 9.34 7.13 y5.66 25.77 – 52 VDB Q qQbf af

71 gravel 93.9 79.3 12.04 9.19 y1.21 64.87 59.42 95 VDB Q qQbf af0.5Ž . Ž .72 log S Q and log D srmrg 92.2 71.4 12.78 8.08 y2.40 92.68 29.44 157 VDB Q qQ10 v 10 50 af bf

73 sand 97.6 63.6 8.52 6.36 y4.93 25.23 70.00 52 VDB Q qQaf bf

74 gravel 90.9 79.3 11.46 8.38 y1.38 59.47 38.33 95 VDB Q qQaf bf

asrmrg refers to sand, mixed, and gravel channels modeled collectively.b Ž .M, B, I, BI, and Q refer to meandering, braided, incising, braided or incising, and quasi-equilibrium channels CEM Types 4 and 5 , respectively. Model parameters are

Ž . 3defined as in Eq. 12 . In single-parameter models, D is in meters. In two-parameter models, D is in millimeters. Discharges are in units of m rs, and slopes are50 50

dimensionless.cQ qQ refers to combining Q and Q with Q used as first priority. Q qQ refers to combining Q and Q with Q used as first priority. VDB refers to theaf bf af bf af bf af bf af bf

Ž .exclusive use of van den Berg’s 1995 data set.


models were most appropriate for rivers with gravelbeds.

3.1. Channels with sand beds

Diagrams based on the single parameter logisticregression models provide a convenient means forenvisioning the risk associated with varying levels of

Žspecific stream power relative to sediment size Fig..1 . In general, systems that are particularly suscepti-

ble to abrupt shifts between the specified channelpatterns produce steep logistic curves. Transitions inchannel type were particularly abrupt in the compari-son of stable meandering and incising channels with

Ž .sand beds Fig. 1 . The logistic curve, representingŽthe transition from meandering to incising model

.15, ns90 , classified stable meandering and incis-ing channels with sand beds with 89.6% and 92.9%accuracy, respectively. The accuracy of predicting anincising channel was improved to over 95% in the

Ž .model using only annual flood discharges Table 1 .

In channels with sand beds, a greater level of flowenergy relative to the caliber of bed materials isassociated with a fully braided form when compared

Ž .to an incised form Fig. 1 . This seems logicalbecause a greater shear stress is typically exerted onthe bed of the channel than on the banks, and onewould suspect that erosion of the bed would oftenprecede bank erosion in noncohesive sand. More-over, incision is a process typically occurring at theinitiation of channel adjustment as opposed to braid-ing that, at least in the case of a formerly meanderingchannel, is a channel form that results after signifi-cant adjustment. The greater magnitude ofŽ .0.5S QrD in a braided channel may, in many50

situations, reflect increases in slope associated withbank erosion and straightening as well as possibledecreases in the median size of bed material. These

Ž .0.5processes could increase the value of S QrD as50

a channel adjusts to the form of a braided channel.The mobility index was also effective in predict-

ing the transition from a stable meandering form toan incising or braiding form of a channel with a

Fig. 1. Results of regression models 2 and 15 comparing stable meandering with braiding and incising streams with sand beds. Discharge isrepresented by annual flood as first priority then bankfull.


0.5 Ž .Fig. 2. Probability of incising or braiding as a function of SQ and D for streams with sand beds model 21, ns109 . Discharge is50

represented by annual flood as first priority then bankfull.

0.5 Ž .Fig. 3. Probability of incising or braiding as a function of S Q and D for streams with sand beds model 22, ns109 . Discharge isv 50

represented by annual flood as first priority then bankfull.


sand bed. The relationship resulting from model 19is:

Qexp 4.51q13.15log S10 (ž /D50

ps 13Ž .Q

1qexp 4.51q13.15log S10 (ž /D50

where S is the dimensionless bedslope; Q is theannual flood or bankfull discharge in m3rs; and D50

Ž .is the median size of particle in meters. Eq. 13classified stable meandering versus incising or braid-ing streams with sand beds with 85.4% and 91.8%

Ž .accuracy, respectively ns109 . The logistic curveŽ .represented by Eq. 13 is essentially equivalent to

the incision curve presented in Fig. 1, with a 50%risk corresponding to a mobility index of 0.45 mrs0.5.Again, this indicates that the incision data define the

Ž .0.5lower limit of S QrD values associated with50

instability of sand bed channels. The results of thetwo-parameter models for predicting an incising orbraiding channel form in sand bed channels are also

presented in Figs. 2 and 3. The predictive accuraciesof the two-parameter models using bedslope areessentially equal to the accuracies of the single pa-rameter models. This reflects the fact that even in thetwo-parameter models channel pattern varied withD0.5. When using valley slope, however, the accura-50

cies of the two-parameter models are superior whencompared to the mobility index models.

Exclusive use of bankfull discharges in conjunc-tion with bedslopes resulted in the most accuratepredictions of braiding in channels with sand beds.Unfortunately, the limited number of annual flooddata from braiding channels precluded any meaning-ful analysis based solely on annual flood data. Mod-els utilizing annual flood data in conjunction with

Ž .bankfull discharge data for missing values ns67were 96% accurate for classifying meandering chan-nels, but only managed to classify braiding channelswith 82% and 68% accuracy using bedslope andvalley slope data, respectively. Accuracy for classifi-cation of braiding channels increased to 89% forbedslope and valley slope models when only bank-

Ž .full values were used ns44 . In contrast, the ex-

0.5 Ž .Fig. 4. Probability of braiding as a function of SQ and D for streams with sand beds model 10, ns67 . Discharge is represented by50

annual flood as first priority then bankfull.


clusive use of bankfull data reduced accuracy forprediction of meandering channels. The models de-veloped with only bankfull discharge data were morestrongly influenced by the D term and suggested50

that very high values of stream power were neces-sary to achieve braiding in coarse sand. These resultswere strongly influenced by a few meandering chan-nels in coarse sand occurring on relatively steepvalleys. The models using annual flood as first prior-ity were based on 23 more samples and were lessinfluenced by these extreme values as evidenced by

2 Ž .higher x values Figs. 4 and 5 .

3.2. Channels with graÕel beds

Braiding in channels with gravel beds apparentlyis more difficult to predict accurately than incising orbraiding in channels with sand beds, given the mod-els examined. Bank resistance, flood plain resistanceto the development of chute cutoffs, the rate ofsediment delivery from eroding banks, and otherfactors may be more influential in controlling theform of gravel bed channels. In many instances, D50

exerted much less influence over the transition frommeandering to braiding in gravel channels as com-pared to sand channels. This is evidenced by therelatively low values of b in some of the two-2

Žparameter logistic regression models for gravel Ta-.ble 1 . These low values indicate that holding the

exponent of the D term at a constant value of 0.5,50

as in the mobility index, reduces the accuracy ofsome models. The best-fitting two-parameter modelssuggest that the stream power associated with acertain risk of a braided channel varies with D to50

the exponent 0.40 to 0.49. In general, regressionrelationships for stable rivers with gravel beds re-ported by other investigators suggest that bed materi-als in the range of D to D exert more control on65 90

Ž .channel slope than D Knighton, 1998 . Unfortu-50

nately, data regarding coarser fractions of bed mate-rials were not available for a large number of streamsused in this study.

The two-parameter models developed using a mixof 127 annual flood and bankfull data classifiedstable meandering channels with over 93% accuracywhen using either bedslope or valley slope. By con-

0.5 Ž .Fig. 5. Probability of braiding as a function of S Q and D for streams with sand beds model 13, ns67 . Discharge is represented byv 50



trast, the same models classified braiding channelswith only 77% and 68% accuracy when using beds-lope and valley slope, respectively. Diagrams depict-ing these models for gravel streams are presented inFigs. 6 and 7. Exclusive use of bankfull dischargedata did not substantially improve the accuracy ofthe two-parameter models.

3.3. Comparisons with the meandering–braiding( )thresholds of Õan den Berg 1995

When compared to the data set compiled by vanŽ .den Berg 1995 , the expanded data set developed

for this study resulted in a 1–10% lower rate ofcorrect classification in the logistic models when

Žusing valley slope as an independent variable Table.1 . Nonetheless, the logistic models extend van den

Berg’s original data set and assign risks of braidingas opposed to a discrete threshold. In contrast to vanden Berg’s combined analysis of streams with D50

between 0.1 and 100 mm, these results suggest thatseparating sand and gravel streams yields more accu-

rate predictions of channel form for each type. At-tempts to develop a robust predictor of braiding forsand and gravel streams, taken collectively, produceda relatively high percentage of misclassified streams.Re-analysis of the data used to develop van denBerg’s discriminant function also indicated that sepa-rating sand and gravel yielded slightly more accurate

Ž .results Table 1 .Part of van den Berg’s success in developing a

single discriminator for the transition from meander-ing to braiding in streams with a D of 0.1–10050

mm may be attributed to the regime widths assumedfor estimating specific stream power. By selecting aregime equation from the design of an irrigationcanal, streams with sand beds were assumed to be57% wider than streams with gravel beds in van denBerg’s analysis. It follows that estimates of specificstream power for sand bed streams were biased 57%lower than the estimates for streams with gravelbeds. In reality, streams with sand beds transportingsubstantial sediment loads may require more specificstream power than threshold channels composed of

Ž .fine to medium gravel Howard, 1980 . van den

0.5 Ž .Fig. 6. Probability of braiding as a function of SQ and D for streams with gravel beds model 11, ns127 . Discharge is represented by50



0.5 Ž .Fig. 7. Probability of braiding as a function of S Q and D for streams with gravel beds model 14, ns127 . Discharge is representedv 50

by annual flood as first priority then bankfull.

Berg’s selection of ws4.7Q0.5 and ws3Q0.5 forthe regime width of sand and gravel channels, re-spectively, would tend to remove this offset in theestimated stream power and facilitate fitting a singlefunction for sand and gravel.

By applying the regime width that van den Bergassumed for gravel channels, his original discrimina-

Žtor based on 28 sand and 98 gravel bed streams Eq.Ž ..1 may be restated as:

0.42'S Q s0.0151 D 14Ž .v 50

where valley slope is dimensionless; Q is the bank-full discharge in m3rs; and D is the median size50

of particle in millimeters. For comparison, the logis-tic regression results for 127 gravel streams utilizingannual flood data in combination with bankfull dis-

Ž .charge model 14, Fig. 7 may be reduced to:

0.43'S Q s0.0153D 15Ž .v 50

Ž .with units corresponding to Eq. 14 . This equationrepresents a 50% probability of braiding and is al-most exactly the same result that van den Bergproposed for D between 0.1 and 100 mm. The50

similarity of these relationships indicates that despite

the inclusion of more data, the 50% risk level forgravel is essentially equivalent to van den Berg’soriginal discriminator. If van den Berg’s data onsand channels are combined with the gravel datawithout assuming a 57% difference in width, theresults are significantly different. In this case, the50% risk of braiding for sand and gravel correspondsto:

0.32'S Q s0.0245D 16Ž .v 50

This constant of 0.0245 indicates a value of S Q0.5v

required for a 50% risk of braiding in 1 mm sandthat is substantially higher than the value 0.0151

Ž . Žsuggested by Eq. 14 van den Berg’s original dis-.criminator , which is biased toward lower powers for

the occurrence of braiding in channels with sandbeds.

In general, logistic models of the transition tobraiding in streams with sand beds differed substan-tially from van den Berg’s original discriminantfunction. This resulted from a few influential data onbraiding channels with D between 1 and 2 mm,50

despite logistic models being developed in this study


being based on 67 streams with sand beds as op-posed to the 28 used by van den Berg. Meanderingchannels in coarse sand exhibited very steep valleyslopes and strongly influenced the nature of theregression relationship. As a result, the predictedlevel of S Q0.5 that is associated with a certain riskv

of braiding is extremely sensitive to D . Until50

further data on streams with beds of coarse sand areobtained, the relationship between S Q0.5 and Dv 50

will remain unclear. The logistic regression modelsresulting from van den Berg’s original data set arepresented in Figs. 8–10.

3.4. Some limitations of the approach

Statistical analysis is a tool for associating obser-vations with measurements of variables that arethought to represent underlying controls. But theassociations between favored variables and observedoutcomes do not necessarily imply causation. Specif-ically, these analyses do not demonstrate how chan-nels develop an incising or braiding form. Instead,the logistic regression models indicate that certain

channel forms are clearly associated with specificcombinations of slope, discharge, and characteristicsof bed materials. Inferences may be drawn fromthese associations, but the models do not reveal thetrajectories of specific stream power, size of bedmaterial, and roughness characteristics as streamsrespond to erosive forces. Coarsening andror armor-ing of beds may be particularly confounding. Inaddition, gravel bed channels with mobility indicessuggesting a low risk of braiding may still be later-ally unstable.

Another limitation of the logistic regression mod-els is the absence of variables representing sedimentsupply and the ratio of bank resistance to bed resis-tance. Channels with sand beds are extremely sensi-tive to the inflowing sediment load delivered fromupstream reaches. Even if a meandering channel witha sand bed has a low risk of incision according to theresults of the logistic regression analysis, it mightstill incise if the inflowing sediment load is substan-tially reduced. The limits of stability presented inthis study might be more accurately portrayed asthresholds of catastrophic channel change. The re-sults should not be interpreted to suggest that inci-

Fig. 8. Logistic regression analysis of van den Berg’s original data set using bankfull discharge and D between 0.1 and 100 mm.50


Fig. 9. Logistic regression analysis of van den Berg’s original data for streams with sand beds using bankfull discharge.

Fig. 10. Logistic regression analysis of van den Berg’s original data for streams with gravel beds using bankfull discharge.


sion or widening will not occur in channels withŽ .0.5sand beds at values of S QrD less than about50

0.3 mrs0.5.The logistic regression models presented herein

should be considered a starting point. The modelsshould be refined for general and regional applica-tion as more data on unstable channels becomeavailable. In particular, regional relationships forchannels with sand beds should be developed in aneffort to include the effects of sediment supply anddiffering bank conditions. Because data on incisionby the stage of channel evolution are rare for areasoutside the Yazoo River watershed of northern Mis-sissippi, the incision data used in this analysis werenecessarily limited to that region. These streams arequite similar and generally have highly cohesivebanks and medium sand beds. Inclusion of otherincision data could have biased estimates of incipientincision because many observations of instability arerecorded after a channel response has resulted in a

Ž .decrease in stream power i.e., slope . An incisingstream that is in CEM stage 3 or 4, for example,typically looks severely unstable. But the process ofchannel incision, having exceeded the critical heightfor bank failure, may have already resulted in asignificant decrease in channel slope. It is likely thatsome streams, that are assumed to be rapidly incis-ing, could have previously lowered values ofŽ .0.5S QrD if slope is measured after incision has50

progressed for an adequate period of time. Thus, itwill be necessary to identify streams with sand bedsby the stage of channel evolution if regional analysesare to accurately reflect levels of stream power thatresult in incision.

4. Conclusions

The high predictive accuracies of the resultingstatistical models suggest that logistic regressionanalysis is an appropriate technique for associatingbasic hydraulic data with various stable and unstablechannel forms. The utility of this approach for identi-fying streams that are particularly susceptible tochanges in the controlling variables is also clear.Traditional discriminators of channel pattern do notadequately portray the fuzziness of these transitions.

In particular, the relative flatness of logistic curvesproduced by the meandering–braiding transition re-flects the omission of key factors that control lateraladjustment processes and the preponderance of inter-mediate channel patterns. The probabilistic diagramsresulting from these analyses provide users with amore realistic assessment of the uncertainty associ-ated with previously identified thresholds of channelinstability. In addition, the mobility indexŽ .0.5S QrD proved highly effective in predicting50

the form of channels with sand beds with limitedinformation. This parameter is also quite useful forpredicting the form of streams with gravel beds andin scaling stream slopes across a broad range ofwatershed scales and geologic conditions.

The logistic regression models suggest that chan-nels with sand beds with cohesive banks may beparticularly susceptible to modest increases in spe-cific stream power. Although incising and braiding,streams with sand beds exhibit similarly high levelsof specific stream power relative to stable meander-ing streams, incision apparently initiates at levels ofspecific stream power that are less than those foundin fully braided channels. For channels with gravelbeds, the logistic regression results provide a proba-

Ž .bilistic version of van den Berg’s 1995 thresholdfor the transition from meandering to braiding. Al-though data on certain size fractions remain sparse,the logistic models suggest that the transition tobraiding in channels with sand beds may be moredependent on the caliber of bed material than inchannels with gravel beds.

In general, methods for systematic evaluation ofpotential instability in alluvial channels at the water-shed scale are still primitive. Such an evaluationmight be performed in a particular basin by estimat-ing available stream power using relations of re-gional flow and a geographic information system todetermine whether excess stream power is available

Ž .to initiate channel instability. Booth 1991 and Si-Ž .mon and Downs 1995 suggest that the likelihood of

channel instability can be assessed in a general man-ner for study sites in a given drainage basin and thatsuch an approach should be pursued. The logisticregression approach developed in this study providesan alternative tool for risk-based management offluvial systems in the context of channel modifica-tions and watershed disturbances.


Acknowledgements

We are very grateful to E.E. Wohl, J.D. Phillips,and an anonymous reviewer for constructive reviewsof the manuscript. R.L. Kelley and P.L. Chapmanprovided invaluable assistance with logistic regres-sion analyses.

References

Begin, Z.B., 1981. The relationship between flow shear stress andstream pattern. J. Hydrol. 52, 307–319.

Booth, D.B., 1991. Urbanization and the natural drainage systemŽ .mpacts, solutions and prognoses. NW Environ. J. 7 1 , 93–

118.Brewer, P.A., Lewin, J., 1998. Planform cyclicity in an unstable

reach: complex fluvial response to environmental change.Earth Surf. Processes Landforms 23, 989–1008.

Bridge, J.S., 1993. The interaction between channel geometry,water flow, sediment transport and deposition in braided rivers.

Ž .In: Best, J.L., Bristow, C.S. Eds. , Braided Rivers. SpecialPublication of the Geological Society of London, vol. 75, pp.13–71.

Bull, W.B., 1979. Threshold of critical power in streams. Geol.Soc. Am. Bull. 90, 453–464.

Carson, M.A., 1984. The meandering-braided river threshold: areappraisal. J. Hydrol. 113, 1402–1421.

Chang, H.H., 1985. River morphology and thresholds. J. Hydraul.Eng. 111, 503–519.

Chitale, S.V., 1973. Theories and relationships of river channelpatterns. J. Hydrol. 19, 285–308.

Christensen, R., 1997. Log-Linear Models and Logistic Regres-sion. Springer-Verlag, New York, 483 pp.

Ferguson, R.I., 1987. Hydraulic and sedimentary controls onŽ .channel pattern. In: Richards, K.S. Ed. , River Channels:

Environment and Process. Blackwell, Oxford, pp. 129–158.Fredsøe, J., 1978. Meandering and braiding of rivers. J. Fluid

Mech. 84, 609–624.Fukuoka, S., 1989. Finite amplitude development of alternate

Ž .bars. In: Ikeda, S., Parker, G. Eds. , River Meandering. Am.Geophys. Union, Water Resources Monographs, vol. 12, pp.237–265.

Galay, V.J., 1983. Causes of river bed degradation. Water Resour.Ž .Res. 19 5 , 1057–1090.

Graf, W.L., 1981. Channel instability in a braided sand-bed river.Water Resour. Res. 17, 1087–1094.

Harvey, M.D., Watson, C.C., 1986. Fluvial processes and morpho-logical thresholds in incised channel restoration. Water Re-

Ž .sour. Bull. 22 3 , 359–368.Howard, A.D., 1980. Thresholds in river regimes. In: Coates,

Ž .D.R., Vitek, J.D. Eds. , Thresholds in Geomorphology.George Allen and Unwin, Boston, MA, pp. 227–258.

Knighton, A.D., 1998. Fluvial Forms and Processes: A NewPerspective. Wiley, New York, 383 pp.

Knighton, A.D., Nanson, G.C., 1993. Anastomosis and the contin-uum of channel pattern. Earth Surf. Processes Landforms 18,613–625.

Lane, E.W., 1957. A study of the shape of channels formed bynatural streams flowing in erodible material. Missouri RiverDivision Sediment Series No. 9. U.S. Army Engineer Divi-sion, Missouri River Corps of Engineers, Omaha, NE.

Leopold, L.B., Wolman, M.G., 1957. River channel patterns:braided, meandering and straight. U.S. Geol. Surv. Prof. Pap.282-B, 85 pp.

Menard, S.W., 1995. Applied Logistic Regression Analysis. SagePublications, Thousand Oaks, CA, 98 pp.

Montgomery, D.R., Buffington, J.M., 1997. Channel reach mor-phology in mountain drainage basins. Geol. Soc. Am. Bull.

Ž .109 5 , 596–611.Neter, J., Wasserman, W., Whitmore, G.A., 1988. Applied Statis-

tics. Allyn and Bacon, Boston, MA, 1006 pp.Osterkamp, W.R., 1978. Gradient, discharge and particle size

relations of alluvial channel in Kansas, with observations onbraiding. Am. J. Sci. 278, 1253–1268.

Parker, G., 1976. On the cause and characteristic scales of mean-dering and braiding in rivers. J. Fluid Mech. 76, 457–480.

SAS Institute, 1990. SASrSTAT User’s Guide, version 6. 4thedn. SAS Institute, Cary, NC, 1686 pp.

Schumm, S.A., 1977. The Fluvial System. Wiley-Interscience,New York, 338 pp.

Schumm, S.A., Khan, H.R., 1972. Experimental study of channelpatterns. Geol. Soc. Am. Bull. 83, 1755–1770.

Schumm, S.A., Harvey, M.D., Watson, C.C., 1984. Incised Chan-nels: Morphology, Dynamics, and Control. Water Resour.Publ., Littleton, CO, 200 pp.

Simon, A., 1989. A model of channel response in disturbedŽ .alluvial channels. Earth Surf. Processes Landforms 14 1 ,

11–26.Simon, A., Downs, P.W., 1995. An interdisciplinary approach to

evaluation of potential instability in alluvial channels. Geo-morphology 12, 215–232.

Simon, A., Hupp, C.R., 1992. Geomorphic and Vegetative Recov-ery Processes along Modified Stream Channels of West Ten-nessee. U.S. Geological Survey Open-File Report 91-502, 142pp.

StatSoftw , 1999. Electronic Statistics Textbook. StatSoft, Tulsa,OK.

Tung, Y., 1985. Channel scouring potential using logistic analysis.Ž .J. Hydraul. Eng. 111 2 , 194–205.

USACE, 1997. HEC-RAS River Analysis System. Version 2.2.U.S. Army Corps of Engineers, Hydrologic Engineering Cen-ter, Davis, CA.

van den Berg, J.H., 1995. Prediction of alluvial channel pattern ofperennial rivers. Geomorphology 12, 259–279.

Waters, T.F., 1995. Sediment in Streams: Sources, BiologicalEffects, and Control. American Fisheries Monograph, vol. 7,American Fisheries Society, Bethesda, MD, 251 pp.

Watson, C.C., Harvey, M.D., Biedenharn, D.S., Combs, P.G.,1988a. Geotechnical and hydraulic stability numbers channelrehabilitation: Part I. The approach. In: Abt, S.R., Gessler, J.Ž .Eds. , Proc. ASCE Hydr. Div. Nat. Conf. ASCE, New York,pp. 20–125.


Watson, C.C., Harvey, M.D., Biedenharn, D.S., Combs, P.G.,1988b. Geotechnical and hydraulic stability numbers channelrehabilitation: Part II. Application. In: Abt, S.R., Gessler, J.Ž .Eds. , Proc. ASCE Hydr. Div. Nat. Conf. ASCE, New York,pp. 126–131.

Watson, C.C., Thornton, C.I., Kozinski, P.B., Bledsoe, B.P., et al.,1998. Demonstration Erosion Control Monitoring Sites 1997Evaluation. Report for USACE, Waterways Experiment Sta-tion, Vicksburg, MS, 129 pp.

Watson, C.C., Bledsoe, B.P., Kozinski, P.B., 1998. YalobushaBasin Geomorphic Assessment. Report for the USACE, Wa-terways Experiment Station, Vicksburg, MS, 64 pp.

Watson, C.C., Bledsoe, B.P., Holmquist-Johnson, C.L., Mooney,

D.M., Haas, A.M., 1999. Demonstration Erosion ControlMonitoring Sites 1999 Evaluation. Report for USACE, Engi-neer Research and Department Center, Vicksburg, MS, 157pp.

Werritty, A., 1997. Short-term changes in channel stability. In:Ž .Thorne, C.R., Hey, R.D., Newson, M.D. Eds. , Applied Flu-

vial Geomorphology for River Engineering and Management.Wiley, Chichester, UK, pp. 47–65.

Wolman, M.G., 1954. A method of sampling coarse bed material.Am. Geophys. Union Trans. 35, 951–956.

Yang, C.T., 1976. Minimum unit stream power and fluvial hy-draulics. J. Hydraul. Div., ASCE, Proc. Pap. 12238, 102,Ž .HY7 , 919–934.

logistic analysis of channel pattern thresholds:...

Documents